Centre of Mass and Collisions

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $2.0\ \mathrm{N\,m^{-1}}$ and the mass of the block is 2.0 kg. Ignore the mass of the spring. Initially the spring is in an unstretched condition. Another block of mass 1.0 kg moving with a speed of $2.0\ \mathrm{m\,s^{-1}}$ collides elastically with the first block. The collision is such that the 2.0 kg block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is ______.

A block of mass $M$ has a circular cut with a frictionless surface as shown. The block rests on the horizontal frictionless surface of a fixed table. Initially the right edge of the block is at $x = 0$, in a co-ordinate system fixed to the table. A point mass $m$ is released from rest at the topmost point of the path as shown and it slides down. When the mass loses contact with the block, its position is $x$ and the velocity is $v$. At that instant, which of the following options is/are correct?

Consider regular polygons with number of sides $n = 3, 4, 5\ldots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\Delta$. Then $\Delta$ depends on $n$ and $h$ as

A rigid uniform bar AB of length $L$ is slipping from its vertical position on a frictionless floor (as shown in the figure). At some instant of time, the angle made by the bar with the vertical is $\theta$. Which of the following statements about its motion is/are correct?

A particle A of mass m and initial velocity $v$ collides with a particle B of mass $\dfrac m2$ which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths $\lambda_A$ to $\lambda_B$ after the collision is :

An object is dropped from a height h from the ground. Every time it hits the ground it looses 50% of its kinetic energy. The total distance covered as $t\to\infty$ is :

Two particles A and B of equal mass M are moving with the same speed $v$ as shown in the figure. They collide completely inelastically and move as a single particle C. The angle θ that the path of C makes with the X-axis is given by :

Two thin circular discs of mass $m$ and $4m$, having radii of $a$ and $2a$, respectively, are rigidly fixed by a massless, rigid rod of length $l=\sqrt{24}\,a$ through their centers. This assembly is laid on a firm and flat surface, and set rolling without slipping on the surface so that the angular speed about the axis of the rod is $\omega$. The angular momentum of the entire assembly about the point 'O' is $\vec L$ (see the figure). Which of the following statement(s) is(are) true?

A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude $A$ about an equilibrium position $x_0$. Consider two cases: (i) when the block is at $x_0$; and (ii) when the block is at $x=x_0+A$. In both the cases, a particle with mass $m\,(<M)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass $m$ is placed on the mass $M$?

In the figure shown ABC is a uniform wire. If centre of mass of wire lies vertically below point A, then $\dfrac{BC}{AB}$ is close to :

A neutron moving with a speed '$v$' makes a head on collision with a stationary hydrogen atom in ground state. The minimum kinetic energy of the neutron for which inelastic collision will take place is :

A fission reaction is given by $^{236}_{92}\mathrm U\to{}^{140}_{54}\mathrm{Xe}+{}^{94}_{38}\mathrm{Sr}+x+y$, where $x$ and $y$ are two particles. Considering $^{236}_{92}\mathrm U$ to be at rest, the kinetic energies of the products are denoted by $K_{Xe}$, $K_{Sr}$, $K_x$ (2 MeV) and $K_y$ (2 MeV), respectively. Let the binding energies per nucleon of $^{236}_{92}\mathrm U$, $^{140}_{54}\mathrm{Xe}$ and $^{94}_{38}\mathrm{Sr}$ be 7.5 MeV, 8.5 MeV and 8.5 MeV, respectively. Considering different conservation laws, the correct option(s) is(are)

A particle of mass m moving in the $x$ direction with speed $2v$ is hit by another particle of mass 2m moving in the $y$ direction with speed $v$. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to :

Distance of the centre of mass of a solid uniform cone from its vertex is $z_0$. If the radius of its base is R and its height is h then $z_0$ is equal to :

A block of mass m = 10 kg rests on a horizontal table. The coefficient of friction between the block and the table is 0.05. When hit by a bullet of mass 50 g moving with speed $v$, that gets embedded in it, the block moves and comes to stop after moving a distance of 2 m on the table.

If a freely falling object were to acquire speed $\dfrac v{10}$ after being dropped from height H, then neglecting energy losses and taking $g=10$ ms$^{-2}$, the value of H is close to :

A block of mass m = 0.1 kg is connected to a spring of unknown spring constant k. It is compressed to a distance $x$ from its equilibrium position and released from rest. After approaching half the distance $\left(\dfrac x2\right)$ from equilibrium position, it hits another block and comes to rest momentarily, while the other block moves with a velocity 3 ms$^{-1}$. The total initial energy of the spring is :

A large number (n) of identical beads, each of mass m and radius r are strung on a thin smooth rigid horizontal rod of length L (L>>r) and are at rest at random positions. The rod is mounted between two rigid supports (see figure). If one of the beads is now given a speed $v$, the average force experienced by each support after a long time is (assume all collisions are elastic) :

A uniform thin rod AB of length L has linear mass density $\mu(x)=a+\dfrac{bx}L$, where $x$ is measured from A. If the CM of the rod lies at a distance of $\left(\dfrac7{12}L\right)$ from A, then a and b are related as :

A tennis ball is dropped on a horizontal smooth surface. It bounces back to its original position after hitting the surface. The force on the ball during the collision is proportional to the length of compression of the ball. Which one of the following sketches describes the variation of its kinetic energy $K$ with time $t$ most appropriately? The figures are only illustrative and not to the scale.

A thin bar of length L has a mass per unit length $\lambda$, that increases linearly with distance from one end. If its total mass is M and its mass per unit length at the lighter end is $\lambda_o$, then the distance of the centre of mass from the lighter end is :